#241282

order	512
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,7)(3,8,4,6)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24)(25,26,29,32)(27,30,34,39)(28,36,37,31)(33,40,35,38);(1,3,5,4)(2,6,7,8)(25,27)(26,30)(28,33)(29,34)(31,38)(32,39)(35,37)(36,40);(1,4,5,3)(2,6,7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21)(27,33)(28,37)(30,38)(31,36)(34,35)(39,40);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148478);